The wires coming off the trim motor are blue / green.
On the other boat I guess they were connected to a 4x6 box labled fuses inside.
That box has out of it, a plug connector with blue / green wires
a plug connector with 2 black wires
a long battery connector cable
a plug conector with blue/white, red/white,
a bare white/brown wire
a bare black/brown wire
Can you tell me how each of these wires should be used?
Also, back to my original question, can they be by-passed our should they
be passed or not?
You didn't mention what motor you are replacing, but it sounds like an older motor with a three wire tilt motor. If that's the case, you have to use the relay set up, because the two wire tilt motor uses green and blue for both up and down. if green positive and blue negative, tilt goes down, if blue pos and green neg goes up. relays allow for this change in polarity.
If you private me Ill send you all the wiring diagrams that you will need. Some are too large to put on this site, and are in pdf format so I cannot use photobucket as a host. The plug conector with blue/white, red/white, green/white wires are for the trim switch connections. Blue is up and green is down in these applications. It sounds though like you are trying to run a 2 wire motor on a 3 wire setup. The wiring diagram that you pic will more easily illustrate which you have and what you have to do to get it running.